/*! \page OBDMDoc one-body density matrix

Keyword: OBDM

\section description Description

Spherically averaged one-body density matrix for up-spin electrons
\f$ \rho_{1\uparrow}^{sph}(|{\bf r}|) \f$, 
\f[
\rho_{1\uparrow}({\bf r}_1,{\bf r}_1')
=\int d^3r_2\ldots d^3r_N\,
\Psi^*({\bf r}_1,{\bf r}_2,\ldots,{\bf r}_N)
\Psi({\bf r}_1',{\bf r}_2,\ldots,{\bf r}_N)\,,
\f]
\f[
\rho_{1\uparrow}^{sph}(|{\bf r}|) = \frac1{4\pi}\int d\Omega_{\bf r}\,
\int d^3r_1 \rho_{1\uparrow}({\bf r}_1+{\bf r},{\bf r}_1)\,,
\f]
calculated on a one-dimensional grid. The many-body wave function \f$
\Psi \f$ in the above definition is assumed normalized to
unity. Normalization of the density matrix is chosen so that
\f$ \rho_{1\uparrow}^{sph}(0)=1 \f$. In homogeneous and isotropic
systems, such as homogeneous electron gas, the one-body density matrix is a
spherically symmetric function of only one distance, and therefore
\f$ \rho_{1\uparrow}^{sph} \equiv \rho_{1\uparrow} \f$. In polarized cases,
\f$ N_{\uparrow}\not = N_{\downarrow} \f$, the density matrix for
down-electrons differs from the matrix for up-electrons. Quantity
\f$ \rho_{1\downarrow}^{sph} \f$ is not (yet) implemented.

\section options Options

\subsection reqopt Required 

None.

\subsection optopt Optional

<table>
<tr><th>Option</th><th>Type</th><th>Default</th><th>Description</th></tr>

<tr><td>CUTOFF</td><td>Float</td>
<td>half of the smallest distance in the simulation cell</td>
<td>Largest distance at which the density matrix is calculated. Values
much larger than the default value have little physical meaning.</td>

<tr><td>NGRID</td><td>Integer</td><td>5</td>
<td>Number of points in the interval [0;CUTOFF] where the density
matrix is calculated. The first point is CUTOFF/NGRID, the last point
is CUTOFF.
</td></tr>

<tr><td>AIP</td><td>Integer</td><td>1</td>
<td>Number of directions for spherical averaging using a Gaussian
quadrature rule. Available are Gaussian rules with 4, 6, 12, 18, 26 and 32
points. Value AIP=1 disables spherical averaging and the vector \f$ \bf
r \f$ then points in the x-direction.
</td></tr>
</table>


*/